{"id":1906,"date":"2021-09-15T14:38:32","date_gmt":"2021-09-15T14:38:32","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1906"},"modified":"2023-12-05T09:23:08","modified_gmt":"2023-12-05T09:23:08","slug":"summary-the-central-limit-theorem-for-sums","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/summary-the-central-limit-theorem-for-sums\/","title":{"raw":"Summary: The Central Limit Theorem for Sums","rendered":"Summary: The Central Limit Theorem for Sums"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The central limit theorem (for sums) states that even if a population distribution is non-normal or the shape is unknown, the shape of the sampling distribution of the sample sums will be approximately normal if the sample size is large enough.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The mean of the sampling distribution of the sample sums is equal to the mean of the population times the sample size.<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">The standard deviation of the sampling distribution of the sample sums is the square root of the sample size times the standard deviation of the population.<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<strong>central limit theorem (for sums):\u00a0<\/strong>given a random variable (RV) with known mean \u03bc and known standard deviation, \u03c3, if the size (n) of the sample is sufficiently large, then [latex]\\sum X \\sim N (n \\mu, \\sqrt{n}\\sigma)[\/latex].\u00a0If the size <em>(n)<\/em> of the sample is sufficiently large, then the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample sums will equal n times the population mean.","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The central limit theorem (for sums) states that even if a population distribution is non-normal or the shape is unknown, the shape of the sampling distribution of the sample sums will be approximately normal if the sample size is large enough.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The mean of the sampling distribution of the sample sums is equal to the mean of the population times the sample size.<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">The standard deviation of the sampling distribution of the sample sums is the square root of the sample size times the standard deviation of the population.<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<p><strong>central limit theorem (for sums):\u00a0<\/strong>given a random variable (RV) with known mean \u03bc and known standard deviation, \u03c3, if the size (n) of the sample is sufficiently large, then [latex]\\sum X \\sim N (n \\mu, \\sqrt{n}\\sigma)[\/latex].\u00a0If the size <em>(n)<\/em> of the sample is sufficiently large, then the distribution of the sample sums will approximate a normal distribution regardless of the shape of the population. The mean of the sample sums will equal n times the population mean.<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1906\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li><strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Introductory Statistics . <strong>Authored by<\/strong>: Barbara Illowsky, Susan Dean. <strong>Provided by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/7-key-terms\">https:\/\/openstax.org\/books\/introductory-statistics\/pages\/7-key-terms<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Introductory Statistics \",\"author\":\"Barbara Illowsky, Susan Dean\",\"organization\":\"Open Stax\",\"url\":\"https:\/\/openstax.org\/books\/introductory-statistics\/pages\/7-key-terms\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Access for free at https:\/\/openstax.org\/books\/introductory-statistics\/pages\/1-introduction\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1906","chapter","type-chapter","status-publish","hentry"],"part":262,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1906","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/users\/169134"}],"version-history":[{"count":3,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1906\/revisions"}],"predecessor-version":[{"id":3719,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1906\/revisions\/3719"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/parts\/262"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapters\/1906\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/media?parent=1906"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=1906"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/contributor?post=1906"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/wp-json\/wp\/v2\/license?post=1906"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}